Quasiperiodic Patterns of the Complex Dimensions of Nonlattice Self-Similar Strings, via the LLL Algorithm

Michel L. Lapidus, Machiel van Frankenhuijsen and Edward K. Voskanian
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Michel L. Lapidus: Department of Mathematics, University of California, Riverside, CA 92521, USA
Machiel van Frankenhuijsen: Department of Mathematics, Utah Valley University, Orem, UT 84058, USA
Edward K. Voskanian: Department of Mathematics and Statistics, The College of New Jersey, Ewing, NJ 08618, USA

Mathematics, 2021, vol. 9, issue 6, 1-35

Abstract: The Lattice String Approximation algorithm (or LSA algorithm) of M. L. Lapidus and M. van Frankenhuijsen is a procedure that approximates the complex dimensions of a nonlattice self-similar fractal string by the complex dimensions of a lattice self-similar fractal string. The implication of this procedure is that the set of complex dimensions of a nonlattice string has a quasiperiodic pattern. Using the LSA algorithm, together with the multiprecision polynomial solver MPSolve which is due to D. A. Bini, G. Fiorentino and L. Robol, we give a new and significantly more powerful presentation of the quasiperiodic patterns of the sets of complex dimensions of nonlattice self-similar fractal strings. The implementation of this algorithm requires a practical method for generating simultaneous Diophantine approximations, which in some cases we can accomplish by the continued fraction process. Otherwise, as was suggested by Lapidus and van Frankenhuijsen, we use the LLL algorithm of A. K. Lenstra, H. W. Lenstra, and L. Lovász.

Keywords: lattice and nonlattice self-similar strings; Diophantine approximation; geometric zeta function; complex dimensions; Dirichlet polynomial; roots of Dirichlet polynomials; lattice case; nonlattice case; lattice string approximation (LSA) algorithm; quasiperiodic structure and patterns; simultaneous Diophantine approximation; LLL algorithm; algorithm of Lenstra, Lenstra and Lovász (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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